Inverse Trigonometric Functions Calculator
Calculate inverse trigonometric functions instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Inverse Trigonometric Functions Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: arcsin(x), arccos(x), arctan(x), arccot(x), arcsec(x), arccsc(x)
Worked example — arcsin = 30 deg | arccos = 60 deg | arctan = 26.565 deg
Formula
arcsin(x), arccos(x), arctan(x), arccot(x), arcsec(x), arccsc(x)
Inverse trigonometric functions return the angle whose trigonometric ratio equals the input value. Each has a specific domain restriction and returns values in a principal range to ensure unique results.
Worked Examples
Example 1: Inverse Trig of 0.5
Problem:Find all six inverse trigonometric function values for input 0.5.
Solution:arcsin(0.5) = 30 degrees (0.523599 rad) - verified: sin(30) = 0.5 arccos(0.5) = 60 degrees (1.047198 rad) - verified: cos(60) = 0.5 arctan(0.5) = 26.565051 degrees (0.463648 rad) - verified: tan(26.565) = 0.5 arccot(0.5) = 63.434949 degrees (1.107149 rad) arcsec(0.5) = undefined (|0.5| < 1) arccsc(0.5) = undefined (|0.5| < 1)
Result:arcsin = 30 deg | arccos = 60 deg | arctan = 26.565 deg
Example 2: Inverse Trig of -1
Problem:Calculate all valid inverse trig values for input -1.
Solution:arcsin(-1) = -90 degrees (-pi/2 rad) arccos(-1) = 180 degrees (pi rad) arctan(-1) = -45 degrees (-pi/4 rad) arccot(-1) = 135 degrees (3pi/4 rad) arcsec(-1) = 180 degrees (pi rad) arccsc(-1) = -90 degrees (-pi/2 rad) Note: arcsin(-x) = -arcsin(x) and arccos(-x) = 180 - arccos(x)
Result:arcsin = -90 deg | arccos = 180 deg | arctan = -45 deg
Frequently Asked Questions
What are inverse trigonometric functions?
Inverse trigonometric functions reverse the operation of the standard trigonometric functions. While sin, cos, and tan take an angle and return a ratio, the inverse functions (arcsin, arccos, arctan) take a ratio and return an angle. For example, if sin(30) = 0.5, then arcsin(0.5) = 30 degrees. These functions are written as arcsin(x), arccos(x), arctan(x), or equivalently as sin-1(x), cos-1(x), tan-1(x). Note that sin-1(x) does NOT mean 1/sin(x); it is a completely different function. Inverse trig functions are essential for finding unknown angles in triangles, physics problems, engineering calculations, and many other applications where the ratio of sides is known but the angle is needed.
What are the domains and ranges of inverse trig functions?
Each inverse trig function has a restricted domain and range to ensure it returns a unique value. Arcsin(x) has domain [-1, 1] and range [-90, 90] degrees ([-pi/2, pi/2] radians). Arccos(x) has domain [-1, 1] and range [0, 180] degrees ([0, pi] radians). Arctan(x) accepts all real numbers and has range (-90, 90) degrees ((-pi/2, pi/2) radians). Arccot(x) accepts all real numbers with range (0, 180) degrees ((0, pi) radians). Arcsec(x) requires |x| >= 1 with range [0, 180] excluding 90 degrees. Arccsc(x) requires |x| >= 1 with range [-90, 90] excluding 0 degrees. These ranges are called principal value branches and represent the standard conventions used in mathematics.
Why do inverse trig functions need restricted ranges?
Inverse trig functions need restricted ranges because the original trig functions are periodic and many-to-one: multiple angles produce the same ratio value. For example, sin(30) = sin(150) = 0.5, and infinitely many other angles also have sine equal to 0.5 (like 30 + 360, 150 + 360, etc.). For arcsin(0.5) to return a single definite answer, we must restrict the output to one interval. The convention chooses ranges that include the most commonly used angles: arcsin uses [-90, 90] to cover one complete period of increasing sine values, arccos uses [0, 180] for one complete period of decreasing cosine values, and arctan uses (-90, 90) for one complete period of increasing tangent values. Without these restrictions, the inverse functions would not be true functions in the mathematical sense.
How is arctan used in real-world applications?
Arctan (inverse tangent) is arguably the most widely used inverse trig function in practical applications. In programming and computer graphics, the atan2(y, x) function (a two-argument variant of arctan) is essential for converting Cartesian coordinates (x, y) to polar coordinates and for calculating rotation angles. In navigation, arctan determines bearing angles from coordinate differences. In physics, arctan calculates the angle of a resultant force vector from its components. In electrical engineering, it determines the phase angle in AC circuits from resistance and reactance values. In machine learning, the arctan function (and its scaled variant, the sigmoid function) serves as an activation function in neural networks. The function is also used in calculus as a standard integral result.
What is the relationship between arcsin and arccos?
Arcsin and arccos have a beautiful complementary relationship: arcsin(x) + arccos(x) = 90 degrees (pi/2 radians) for all x in [-1, 1]. This means if arcsin(0.5) = 30 degrees, then arccos(0.5) must equal 60 degrees, and indeed 30 + 60 = 90. This relationship follows directly from the cofunction identity sin(x) = cos(90 - x). Another way to see it: in a right triangle, if one acute angle has sine equal to 0.5, the other acute angle must have cosine equal to 0.5, and the two acute angles sum to 90 degrees. This identity is useful for simplifying expressions and for checking calculations. A similar complementary relationship exists for arctan and arccot: arctan(x) + arccot(x) = 90 degrees for all real x.
What are the derivatives of inverse trig functions?
The derivatives of inverse trig functions are important results in calculus. The derivative of arcsin(x) is 1/sqrt(1-x2), valid for |x| < 1. The derivative of arccos(x) is -1/sqrt(1-x2), which is simply the negative of arcsin's derivative. The derivative of arctan(x) is 1/(1+x2), valid for all real x. The derivative of arccot(x) is -1/(1+x2). The derivative of arcsec(x) is 1/(|x|sqrt(x2-1)), and arccsc(x) has derivative -1/(|x|sqrt(x2-1)). These formulas are derived using implicit differentiation: if y = arcsin(x), then x = sin(y), differentiating gives 1 = cos(y) dy/dx, so dy/dx = 1/cos(y) = 1/sqrt(1-sin2(y)) = 1/sqrt(1-x2). These derivatives appear frequently as integration results.
How do you evaluate inverse trig functions of common values?
For common values, memorize the standard angle results. For arcsin: arcsin(0) = 0, arcsin(1/2) = 30, arcsin(sqrt(2)/2) = 45, arcsin(sqrt(3)/2) = 60, arcsin(1) = 90 degrees. For arccos: arccos(1) = 0, arccos(sqrt(3)/2) = 30, arccos(sqrt(2)/2) = 45, arccos(1/2) = 60, arccos(0) = 90 degrees. For arctan: arctan(0) = 0, arctan(1/sqrt(3)) = 30, arctan(1) = 45, arctan(sqrt(3)) = 60 degrees. Negative inputs flip the sign for arcsin and arctan (odd functions), while for arccos you subtract from 180 degrees: arccos(-x) = 180 - arccos(x). These values come directly from the well-known 30-60-90 and 45-45-90 triangle ratios.
What is the difference between arctan and atan2?
The standard arctan(y/x) function returns values only in the range (-90, 90) degrees and cannot distinguish between angles in different quadrants when x and y have various sign combinations. The atan2(y, x) function solves this by taking two separate arguments and returning the full angle in the range (-180, 180] degrees, correctly identifying all four quadrants. For example, arctan(-1/-1) = arctan(1) = 45 degrees, but atan2(-1, -1) = -135 degrees, correctly placing the angle in the third quadrant. The atan2 function also handles the case where x = 0 (returning 90 or -90 degrees), while arctan(y/0) would be undefined. Most programming languages and scientific software provide atan2 for this reason, and it is the preferred function for coordinate conversions.
How are inverse trig functions used in integration?
Inverse trig functions appear as results of many standard integrals in calculus. The integral of 1/sqrt(1-x2) dx is arcsin(x) + C. The integral of 1/(1+x2) dx is arctan(x) + C. The integral of 1/(x sqrt(x2-1)) dx is arcsec(|x|) + C. More generally, the integral of 1/sqrt(a2-x2) dx is arcsin(x/a) + C, and the integral of 1/(a2+x2) dx is (1/a)arctan(x/a) + C. These patterns are recognized through trigonometric substitution: when you see sqrt(1-x2), substitute x = sin(theta); when you see 1+x2, substitute x = tan(theta). Recognizing these integral forms is a fundamental skill in calculus and appears extensively in physics, engineering, and probability theory.
Can inverse trig functions handle complex numbers?
Yes, inverse trig functions can be extended to complex numbers, and doing so removes the domain restrictions that exist for real numbers. For example, arcsin(2) is undefined for real numbers because no real angle has a sine of 2, but in the complex plane, arcsin(2) = pi/2 - i ln(2 + sqrt(3)) which is approximately 1.5708 - 1.3170i. The complex extensions are defined using logarithms: arcsin(z) = -i ln(iz + sqrt(1-z2)) and arccos(z) = -i ln(z + sqrt(z2-1)). These complex inverse trig functions are used in advanced mathematics, signal processing, and certain physics applications involving wave propagation in absorbing media. For Inverse Trigonometric Functions Calculator, we focus on real-valued inputs and flag values outside the valid real domain as undefined.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Trigonometric Functions Calculator
Calculate trigonometric functions with inputs, formulas, and instant results.
🧮Hyperbolic Functions Calculator
Calculate hyperbolic functions with inputs, formulas, and instant results.
🧮Inverse Function Calculator
Calculate inverse function with inputs, formulas, and instant results.
🧮Inverse Variation Calculator
Calculate inverse variation with inputs, formulas, and instant results.
🧮Inverse Modulo Calculator
Calculate inverse modulo with inputs, formulas, and instant results.
🧮Multiplicative Inverse Calculator
Calculate multiplicative inverse with inputs, formulas, and instant results.
🧮Multiplicative Inverse Modulo Calculator
Calculate multiplicative inverse modulo with inputs, formulas, and instant results.
🧮Inverse Of4x4matrix Calculator
Calculate inverse of4x4matrix with inputs, formulas, and instant results.